This documentation is automatically generated by online-judge-tools/verification-helper
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// verification-helper: PROBLEM https://yukicoder.me/problems/7107 #include<bits/stdc++.h> using namespace std; #define call_from_test #include "../../mod/mint.cpp" #include "../../math/moebius.cpp" #include "../../convolution/divisor.cpp" #undef call_from_test signed main(){ cin.tie(0); ios::sync_with_stdio(0); int n; cin>>n; using M = Mint<int, 998244353>; vector<M> emp(n+1,0),way(n+1,0); M cnt{0}; M all{0}; auto sign=moebius(n); using ll = long long; for(int i=1;i<=n;i++){ emp[i]=M((ll)sign[i])*M(1).pow(n/i); way[i]=M((ll)sign[i])*M(2).pow(n/i); cnt+=emp[i]; all+=way[i]; } LCMConvolution::zeta(way); LCMConvolution::zeta(emp); vector<M> dp0(n+1,0),dp1(n+1,0),dp2(n+1,0); for(int i=1;i<=n;i++){ dp0[i]=emp[i]*emp[i]; dp1[i]=emp[i]*way[i]; dp2[i]=way[i]*way[i]; } LCMConvolution::moebius(dp0); LCMConvolution::moebius(dp1); LCMConvolution::moebius(dp2); M ans=(all-cnt)*(all-cnt); M cof=M(3)/M(4); for(int i=1;i<=n;i++){ ans+=dp0[i]; ans-=dp1[i]*M(2); ans+=dp2[i]*cof.pow(n/i); ans-=dp0[i]; ans+=dp1[i]*M(2); ans-=dp2[i]; } cout<<ans<<endl; return 0; }
#line 1 "test/yukicoder/7107.test.cpp" // verification-helper: PROBLEM https://yukicoder.me/problems/7107 #include<bits/stdc++.h> using namespace std; #define call_from_test #line 1 "mod/mint.cpp" #line 3 "mod/mint.cpp" using namespace std; #endif //BEGIN CUT HERE template<typename T, T MOD = 1000000007> struct Mint{ inline static constexpr T mod = MOD; T v; Mint():v(0){} Mint(signed v):v(v){} Mint(long long t){v=t%MOD;if(v<0) v+=MOD;} Mint pow(long long k){ Mint res(1),tmp(v); while(k){ if(k&1) res*=tmp; tmp*=tmp; k>>=1; } return res; } static Mint add_identity(){return Mint(0);} static Mint mul_identity(){return Mint(1);} Mint inv(){return pow(MOD-2);} Mint& operator+=(Mint a){v+=a.v;if(v>=MOD)v-=MOD;return *this;} Mint& operator-=(Mint a){v+=MOD-a.v;if(v>=MOD)v-=MOD;return *this;} Mint& operator*=(Mint a){v=1LL*v*a.v%MOD;return *this;} Mint& operator/=(Mint a){return (*this)*=a.inv();} Mint operator+(Mint a) const{return Mint(v)+=a;} Mint operator-(Mint a) const{return Mint(v)-=a;} Mint operator*(Mint a) const{return Mint(v)*=a;} Mint operator/(Mint a) const{return Mint(v)/=a;} Mint operator+() const{return *this;} Mint operator-() const{return v?Mint(MOD-v):Mint(v);} bool operator==(const Mint a)const{return v==a.v;} bool operator!=(const Mint a)const{return v!=a.v;} static Mint comb(long long n,int k){ Mint num(1),dom(1); for(int i=0;i<k;i++){ num*=Mint(n-i); dom*=Mint(i+1); } return num/dom; } }; template<typename T, T MOD> ostream& operator<<(ostream &os,Mint<T, MOD> m){os<<m.v;return os;} //END CUT HERE #ifndef call_from_test signed main(){ return 0; } #endif #line 1 "math/moebius.cpp" #line 3 "math/moebius.cpp" using namespace std; #endif //BEGIN CUT HERE // [0, n] vector<int> moebius(int n){ n++; vector<int> pr(n),sq(n); using ll = long long; for(ll i=2;i<n;i++){ if(pr[i]) continue; for(ll j=i;j<n;j+=i) pr[j]=i; for(ll j=i*i;j<n;j+=i*i) sq[j]=1; } vector<int> sign(n,0); sign[1]=1; for(ll i=2;i<n;i++){ if(sq[i]) continue; sign[i]=-sign[i/pr[i]]; } return sign; } //END CUT HERE #ifndef call_from_test //INSERT ABOVE HERE signed main(){ return 0; } #endif #line 1 "convolution/divisor.cpp" #line 3 "convolution/divisor.cpp" using namespace std; #endif // https://noshi91.hatenablog.com/entry/2019/09/23/002445 //BEGIN CUT HERE // O(n \log \log n) namespace DivisorTransform{ template<typename T, typename F> void inc(vector<T> &as,F f){ assert(as[0]==T(0)); int n=as.size(); vector<bool> sieve(n,false); for(int p=2;p<n;p++){ if(sieve[p]) continue; for(int k=1;k*p<n;k++){ sieve[k*p]=true; f(as[k],as[k*p]); } } } template<typename T, typename F> void dec(vector<T> &as,F f){ assert(as[0]==T(0)); int n=as.size(); vector<bool> sieve(n,false); for(int p=2;p<n;p++){ if(sieve[p]) continue; for(int k=(n-1)/p;k!=0;--k){ sieve[k*p]=true; f(as[k],as[k*p]); } } } } namespace GCDConvolution{ template<typename T> void zeta(vector<T> &as){ auto f=[](T &lo,T &hi){lo+=hi;}; DivisorTransform::dec(as,f); } template<typename T> void moebius(vector<T> &as){ auto f=[](T &lo,T &hi){lo-=hi;}; DivisorTransform::inc(as,f); } } namespace LCMConvolution{ template<typename T> void zeta(vector<T> &as){ auto f=[](T &lo,T &hi){hi+=lo;}; DivisorTransform::inc(as,f); } template<typename T> void moebius(vector<T> &as){ auto f=[](T &lo,T &hi){hi-=lo;}; DivisorTransform::dec(as,f); } } //END CUT HERE #ifndef call_from_test //INSERT ABOVE HERE signed main(){ return 0; } #endif #line 10 "test/yukicoder/7107.test.cpp" #undef call_from_test signed main(){ cin.tie(0); ios::sync_with_stdio(0); int n; cin>>n; using M = Mint<int, 998244353>; vector<M> emp(n+1,0),way(n+1,0); M cnt{0}; M all{0}; auto sign=moebius(n); using ll = long long; for(int i=1;i<=n;i++){ emp[i]=M((ll)sign[i])*M(1).pow(n/i); way[i]=M((ll)sign[i])*M(2).pow(n/i); cnt+=emp[i]; all+=way[i]; } LCMConvolution::zeta(way); LCMConvolution::zeta(emp); vector<M> dp0(n+1,0),dp1(n+1,0),dp2(n+1,0); for(int i=1;i<=n;i++){ dp0[i]=emp[i]*emp[i]; dp1[i]=emp[i]*way[i]; dp2[i]=way[i]*way[i]; } LCMConvolution::moebius(dp0); LCMConvolution::moebius(dp1); LCMConvolution::moebius(dp2); M ans=(all-cnt)*(all-cnt); M cof=M(3)/M(4); for(int i=1;i<=n;i++){ ans+=dp0[i]; ans-=dp1[i]*M(2); ans+=dp2[i]*cof.pow(n/i); ans-=dp0[i]; ans+=dp1[i]*M(2); ans-=dp2[i]; } cout<<ans<<endl; return 0; }